I used to know which square-taper bottom brackets were symmetrical and which weren't. Here are the measurements I'd recorded for standard Shimano (UN-5X or UN-7X) cartridge bottom brackets:
107mm: 20.5mm drive-side, 19.5mm non-drive-side. Drive-side 1mm longer
110mm: 20.5mm drive-side, 22.5mm non-drive-side (or, 20 and 22). Non-drive-side 2mm longer

I think 113mm was near-symmetrical, around 23mm for drive-side and non-drive-side.

So, the "worst-case scenario" appears to be true: with a 113mm unit, your crankarm will be 3mm further away from the frame than it was with the 110mm unit.
If you want the crank to be in the same place as with a 110mm bottom bracket, you should get a 107mm or 110mm cartridge to replace. But maybe that 3mm isn't a problem. Your call.